Power graphs have been shown to capture many algebraic properties of a group. For example, the power graph of a finite group is a tree if and only if every element of the group is its own inverse [1]}. If a finite group's power graph lack particular kinds of subgraphs, then its isomorphism types can be classified [2]}. A finite group is a non-cyclic group of prime exponent if and only if its power graph is non-complete and minimally edge-connected [3]}. For more results of this type, see [4]}.
